### 10.1 The binary’s multipole moments

The general expressions of the source multipole moments given by Theorem 6, Equations (85), are first to be worked out explicitly for general fluid systems at the 3PN order. For this computation one uses the formula (91), and we insert the 3PN metric coefficients (in harmonic coordinates) expressed in Equations (115) by means of the retarded-type elementary potentials (117, 118, 119). Then we specialize each of the (quite numerous) terms to the case of point-particle binaries by inserting, for the matter stress-energy tensor , the standard expression made out of Dirac delta-functions. The infinite self-field of point-particles is removed by means of the Hadamard regularization; and dimensional regularization is used to compute the few ambiguity parameters (see Section 8). This computation has been performed in [49] at the 1PN order, and in [33] at the 2PN order; we report below the most accurate 3PN results obtained in Refs. [45443132].

The difficult part of the analysis is to find the closed-form expressions, fully explicit in terms of the particle’s positions and velocities, of many non-linear integrals. We refer to [45] for full details; nevertheless, let us give a few examples of the type of technical formulas that are employed in this calculation. Typically we have to compute some integrals like

where and . When and , this integral is perfectly well-defined (recall that the finite part deals with the bound at infinity). When or , our basic ansatz is that we apply the definition of the Hadamard partie finie provided by Equation (124). Two examples of closed-form formulas that we get, which do not necessitate the Hadamard partie finie, are (quadrupole case )
We denote for example (and ); the constant is the one pertaining to the finite-part process (see Equation (36)). One example where the integral diverges at the location of the particle 1 is
where is the Hadamard-regularization constant introduced in Equation (124).

The crucial input of the computation of the flux at the 3PN order is the mass quadrupole moment , since this moment necessitates the full 3PN precision. The result of Ref. [45] for this moment (in the case of circular orbits) is

where we pose and . The third term is the 2.5PN radiation-reaction term, which does not contribute to the energy flux for circular orbits. The two important coefficients are and , whose expressions through 3PN order are
These expressions are valid in harmonic coordinates via the post-Newtonian parameter given by Equation (188). As we see, there are two types of logarithms in the moment: One type involves the length scale related by Equation (184) to the two gauge constants and present in the 3PN equations of motion; the other type contains the different length scale coming from the general formalism of Part A - indeed, recall that there is a operator in front of the source multipole moments in Theorem 6. As we know, that is pure gauge; it will disappear from our physical results at the end. On the other hand, we have remarked that the multipole expansion outside a general post-Newtonian source is actually free of , since the ’s present in the two terms of Equation (67) cancel out. We shall indeed find that the constants present in Equations (220) are compensated by similar constants coming from the non-linear wave “tails of tails”. Finally, the constants , , and are the Hadamard-regularization ambiguity parameters which take the values (136).

Besides the 3PN mass quadrupole (219, 220), we need also the mass octupole moment and current quadrupole moment , both of them at the 2PN order; these are given by [45]

Also needed are the 1PN mass -pole, 1PN current -pole (octupole), Newtonian mass -pole and Newtonian current -pole:

These results permit one to control what can be called the “instantaneous” part, say , of the total energy flux, by which we mean that part of the flux that is generated solely by the source multipole moments, i.e. not counting the “non-instantaneous” tail integrals. The instantaneous flux is defined by the replacement into the general expression of given by Equation (60) of all the radiative moments and by the corresponding (th time derivatives of the) source moments and . Actually, we prefer to define by means of the intermediate moments and . Up to the 3.5PN order we have

The time derivatives of the source moments (219, 220, 221, 222) are computed by means of the circular-orbit equations of motion given by Equation (189, 190); then we substitute them into Equation (223). The net result is
The Newtonian approximation, , is the prediction of the Einstein quadrupole formula (4), as computed by Landau and Lifchitz [153]. In Equation (224), we have replaced the Hadamard regularization ambiguity parameters and arising at the 3PN order by their values (135) and (137).